I Googled both words (hayah) and (eheye). It turns out that both are used depending on which translation you are reading. Also, there is a footnote in the New Living Translation using either phrase: (I AM WHO I AM) or (I WILL BE WHAT I WILL BE).

I understood His name to mean that He exists. People have never seen God since He is spirit. Moses was sent by God to deliver the Jewish people from Egypt. God would do His wonders through this man. I am sure that He did not want Moses to take the credit for those miracles. The people needed to know that there is an invisible God in heaven and not a man performing such wonders. But, of course this is my understanding. Perhaps, you are correct in your translation.